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F范数度量下的鲁棒张量低维表征

王肖锋 石乐岩 杨璐 刘军 周海波

王肖锋, 石乐岩, 杨璐, 刘军, 周海波. F范数度量下的鲁棒张量低维表征. 自动化学报, 2021, 48(x): 1−14 doi: 10.16383/j.aas.c210375
引用本文: 王肖锋, 石乐岩, 杨璐, 刘军, 周海波. F范数度量下的鲁棒张量低维表征. 自动化学报, 2021, 48(x): 1−14 doi: 10.16383/j.aas.c210375
Wang Xiao-Feng, Shi Le-Yan, Yang-Lu, Liu Jun, Zhou Hai-Bo. Low dimensional representation of robust tensor under F-norm metric. Acta Automatica Sinica, 2021, 48(x): 1−14 doi: 10.16383/j.aas.c210375
Citation: Wang Xiao-Feng, Shi Le-Yan, Yang-Lu, Liu Jun, Zhou Hai-Bo. Low dimensional representation of robust tensor under F-norm metric. Acta Automatica Sinica, 2021, 48(x): 1−14 doi: 10.16383/j.aas.c210375

F范数度量下的鲁棒张量低维表征

doi: 10.16383/j.aas.c210375
基金项目: 国家重点研发计划 (2017YFB1303304), 天津市科技计划重大专项 (17ZXZNGX00110), 国家自然科学基金项目 (52005370)资助
详细信息
    作者简介:

    王肖锋:博士,天津理工大学机械工程学院副教授. 2018年获得河北工业大学工学博士学位.主要研究方向为发育机器人、模式识别与机器学习. E-mail: wangxiaofeng@tjut.edu.cn

    石乐岩:天津理工大学机械工程学院硕士研究生. 2020年获得天津理工大学机械工程学院学士学位.主要研究方向为数据降维和机器学习. E-mail: 691125398@qq.com

    杨璐:博士,天津理工大学机械工程学院副教授,2011年获得吉林大学工学博士学位,主要研究方向为计算机视觉与模式识别. 本文通信作者. E-mail: yanglu8206@163.com

    刘军:博士,天津理工大学机械工程学院教授. 2002年获得日本名古屋大学工学博士学位.主要研究方向为转子故障信号的特征提取与分类识别. E-mail: liujunjp@tjut.edu.cn

    周海波:博士,天津理工大学机械工程学院教授,2005年获得吉林大学工学博士学位,主要研究方向为机器人技术,图像处理和机器视觉,人工智能.E-mail: zhouhaibo@tjut.edu.cn

Low Dimensional Representation of Robust Tensor under F-norm Metric

Funds: Supported by National Natural Science Foundation of China (61773406, 61725306, 61290325), National Major Scientific Research Equipment of China (61927803), Central South University Post-Graduate Independent Exploration and Innovation Project (2021zzts0183), and Hunan Provincial Innovation Foundation for Postgraduate (CX20210242)
More Information
    Author Bio:

    WANG Xiao-Feng Ph. D., an associate professor at the School of Mechanical Engineering, Tianjin University of Technology. He received his Ph. D. degree from Heibei university of Technology in 2018. His research interest covers developmental robotics, pattern recognition and machine learning

    SHI Le-Yan Master student at the School of Mechanical Engineering, Tianjin University of Technology. He received his bachelor degree from Tianjin University of Technology in 2020. His research interest includes dimensionality reduction and machine learning

    YANG Lu Ph. D., an associate professor at the School of Mechanical Engineering, Tianjin University of Technology. She received a Ph.D. (Eng.) from Jilin University in 2011. Her current research interests include computer vision and pattern recognition. Corresponding author of this paper

    LIU Jun    Ph. D., professor at the School of Mechanical Engineer, Tianjin University of Technology. He received his Ph. D. degree from Nagoya university, Japan in 2002. His research field is feature extraction and recognition for rotor fault

    ZHOU Hai-Bo Ph. D., professor at the School of Mechanical Engineer, Tianjin University of Technology. He received his Ph. D. degree from Jilin University, China in 2005. His current research interests include intelligent robot technology, image processing and machine vision, artificial intelligence

  • 摘要: 张量主成分分析(Tensor principle component analysis, TPCA)在彩色图像低维表征领域得到广泛深入研究, 采用$\textit{F}$范数平方作为低维投影的距离度量方式, 表征含离群数据和噪声图像的鲁棒性较弱. $\textit{L}_{1}$范数能够抑制噪声的影响, 但所获的低维投影数据缺乏重构误差约束, 其局部表征能力也较弱. 针对上述问题, 本文利用$\textit{F}$范数作为目标函数的距离度量方式, 提出一种基于$\textit{F}$范数的分块张量主成分分析算法(Block TPCA with $\textit{F}$-norm, BlockTPCA-$\textit{F}$), 提高张量低维表征的鲁棒性. 考虑到同时约束投影距离与重构误差, 提出一种基于比例$\textit{F}$范数的分块张量主成分分析算法(Block TPCA with proportional $\textit{F}$-norm, BlockTPCA-P$\textit{F}$), 其最大化投影距离与最小化重构误差均得到了优化. 然后, 给出了其贪婪的求解算法, 并对其收敛性进行了理论证明. 最后, 对包含不同噪声块和具有实际遮挡的彩色人脸数据集进行实验, 结果表明, 本文所提算法在平均重构误差、图像重构与分类率等方面均得到了明显提升, 在张量低维表征中具有较强的鲁棒性.
    1)  收稿日期 2021-05-10 录用日期 2021-11-02 Manuscript received May 10, 2021; accepted November 2, 2021 国家重点研发计划 (2017YFB1303304), 天津市科技计划重大专项 (17ZXZNGX00110), 国家自然科学基金项目 (52005370) 资助 Supported by National Natural Science Foundation of China(61773406, 61725306, 61290325), National Major Scientific Research Equipment of China (61927803), Central South University Post-Graduate Independent Exploration and InnovationProject (2021zzts0183), and Hunan Provincial Innovation Foundation for Postgraduate (CX20210242) 本文责任编委 Recommended by Associate Editor 1. 天津理工大学天津市先进机电系统设计与智能控制重点实验室 天津 300384 2. 天津理工大学机电工程国家级实验教学示范中心 天津 300384
    2)  1.  Tianjin Key Laboratory for Advanced Mechatronical Sys-tem Design and Intelligent Control, Tianjin University of Tech-nology, Tianjin 300384 2. National Demonstration Center forExperimental Mechanical and Electrical Engineering Education,Tianjin University of Technology, Tianjin 300384
  • 图  1  BlockTPCA-F算法

    Fig.  1  BlockTPCA-F algorithm

    图  2  BlockTPCA-PF算法

    Fig.  2  BlockTPCA-PF algorithm

    图  3  ${{\cal{X}}_m}$, ${{\cal{Y}}_m}$${{\cal{E}}_m}$之间的关系

    Fig.  3  The relation between ${{\cal{X}}_m}$, ${{\cal{Y}}_m}$ and ${{\cal{E}}_m}$

    图  4  GT彩色人脸数据集样本

    Fig.  4  GT color face dataset samples

    图  5  Aberdeen彩色人脸数据集样本

    Fig.  5  Aberdeen color face dataset samples

    图  6  AR彩色人脸数据集样本

    Fig.  6  AR color face dataset samples

    图  7  平均重构误差

    Fig.  7  Average reconstruction error

    图  8  AR数据集下的平均重构误差

    Fig.  8  Average reconstruction error under AR dataset

    图  9  20%噪声下的重构图像

    Fig.  9  Reconstruction images under 20% noise

    图  11  60%噪声下的重构图像

    Fig.  11  Reconstruction images under 60% noise

    图  10  40%噪声下的重构图像

    Fig.  10  Reconstruction images under 40% noise

    表  1  20%噪声下最优平均分类率

    Table  1  Optimal average classification rate under 20% noise

    NPC MPCA TPCA-$L _{1} $-G TPCA-$L _{1} $-NG TPCA-F BlockTPCA-F BlockTPCA-PF
    AB 10 0.9048 0.8974 0.9079 0.9153 0.9153 0.9143
    20 >0.9132 >0.9111 >0.9132 >0.9101 >0.9164 0.9175
    30 0.9090 0.9069 0.9090 0.9069 0.9090 0.9101
    40 0.9058 0.9048 0.9058 0.9026 0.9090 0.9090
    50 0.9048 0.9058 0.9058 0.9005 0.9079 0.9058
    GT 10 0.6940 0.7020 0.7055 0.7055 0.6900 0.6915
    20 0.7015 0.6935 0.6935 0.6950 0.7070 0.7090
    30 0.7005 0.6875 0.6880 0.6905 0.7020 0.7035
    40 0.6900 0.6860 0.6850 0.6845 0.7010 0.7035
    50 0.6855 0.6820 0.6850 0.6840 0.6970 0.7000
    下载: 导出CSV

    表  3  60%噪声下最优平均分类率

    Table  3  Optimal average classification rate under 60% noise

    NPC MPCA TPCA-$L _{1} $-G TPCA-$L _{1} $-NG TPCA-F BlockTPCA-F BlockTPCA-PF
    AB 10 0.7958 0.7958 0.7937 0.7915 0.8148 0.8116
    20 0.7810 0.7810 0.7788 0.7779 0.7926 0.7947
    30 0.7810 0.7746 0.7820 0.7757 0.7788 0.7799
    40 0.7841 0.7746 0.7767 0.7799 0.7778 0.7799
    50 0.7778 0.7757 0.7778 0.7799 0.7757 0.7757
    GT 10 0.5354 0.5550 0.5690 0.5700 0.5690 0.5680
    20 0.5344 0.5450 0.5665 0.5680 0.5580 0.5580
    30 0.5238 0.5455 0.5590 0.5590 0.5510 0.5520
    40 0.5101 0.5435 0.5470 0.5470 0.5500 0.5510
    50 0.5048 0.5405 0.5450 0.5455 0.5470 0.5485
    下载: 导出CSV

    表  4  AR人脸数据集最优平均分类率

    Table  4  Optimal average classification rate of AR face dataset

    NPC MPCA TPCA-$L _{1} $-G TPCA-$L _{1} $-NG TPCA-F BlockTPCA-F BlockTPCA-PF
    AR 10 0.7692 0.7653 0.7692 0.7731 0.8077 0.8077
    20 0.7654 0.7653 0.7654 0.7616 0.8001 0.8001
    30 0.7654 0.7615 0.7653 0.7654 0.8039 0.8039
    40 0.7654 0.7692 0.7692 0.7654 0.8038 0.8038
    50 0.7692 0.7615 0.7654 0.7692 0.8039 0.8039
    下载: 导出CSV

    表  2  40%噪声下最优平均分类率

    Table  2  Optimal average classification rate under 40% noise

    NPC MPCA TPCA-$L _{1} $-G TPCA-$L _{1} $-NG TPCA-F BlockTPCA-F BlockTPCA-PF
    AB 10 0.8804 0.8794 0.8847 0.8772 0.8889 0.8889
    20 0.8783 0.8772 0.8751 0.8709 0.8889 0.8889
    30 0.8624 0.8571 0.8593 0.8635 0.8751 0.8730
    40 0.8519 0.8497 0.8508 0.8614 0.8603 0.8571
    50 0.8497 0.8476 0.8466 0.8519 0.8529 0.8497
    GT 10 0.6243 0.6645 0.6650 0.6630 0.6690 0.6690
    20 0.5788 0.6335 0.6300 0.6295 0.6590 0.6590
    30 0.5556 0.6115 0.6115 0.6115 0.6455 0.6455
    40 0.5471 0.6050 0.6070 0.6070 0.6320 0.6320
    50 0.5439 0.6035 0.6070 0.6065 0.6220 0.6220
    下载: 导出CSV
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  • 收稿日期:  2021-05-10
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